Binping XiaoCollider-Accelerator Department, Brookhaven National Lab

Based on the Ph.D dissertation supervised by C. E. Reece and M. J. Kelley

College of William and Mary, Jefferson Lab

How Good Can an SRF Cavity Be?A New First-Principle Calculation of Field-Dependent RF

Surface Impedance of BCS Superconductor

Thin films and new ideas for SRF,

10/7/2014

1

Outline

2

Cavity performance

BCS theory, Mattis-Bardeen (M-B) theory

and extension

Calculation results and explanation

Theory vs. experiments

Summary

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Physical Review 108,

1175 (1957)

D. C. Mattis and J. Bardeen, Physical Review 111, 412 (1958)

B.P. Xiao, C.E. Reece, M.J. Kelley, Physica C 490 (2013) 26–31

RF surface impedance in cavity

3

A simplified diagram of SRF device.

Introduction to SRF Linac Technology by Rong-li Geng

http://en.wikipedia.org/wiki/Superconducting_Radio_Freque

ncy

current

d

a

a

Zs = Rs + iXs

Cavity Performance

4

H. Pandamsee: “Two Major Open Physics Topic in RF Superconductivity”

1E+09

1E+10

1E+11

0 50 100 150 200

Qu

ali

ty F

acto

r

Peak Magnetic Field [mT]

1.5 GHz 7-cell CEBAF

cavity with 230 μm BCP

230 μm BCP + 34 μm

EP

1.5GHz single cell

CEBAF cavity with 3 h

1400 C baking

All measured at 2.0K

Superheating

Field

• P. Dhakal, G.

Ciovati, G. R.

Myneni, in: IPAC

2012.

• C. E. Reece et al.,

in PAC2005

• C. E. Reece, and

H. Tian, in

LINAC2010

BCS, M-B and extension

5

Each Cooper pair was assumed to have zero net momentum. By minimizing

the free energy one get a single particle distribution function f and Density of

State (DoS), represented by h.

Considering that a single particle scattering from one energy state to

another one, by considering the energy of each state (with one photon added

to either the initial state or the final state.), the probability of initial state and

the possibility of scattering, a scattering form was calculated using f and h.

This scattering form was then applied to the anomalous skin effect to get the

surface impedance.

We assume that all Cooper pairs move with the same velocity Vs, and apply

this assumption to BCS theory to get new f and h.*

These new f and h are applied to get a new scattering form, thus a new

surface impedance.

We average these effects over both RF cycle and depth into the surface to get

the effective surface resistances (Rs) under different fields.**

*J. Bardeen, Reviews of Modern Physics 34 (1962) 667.**I.O. Kulik, V. Palmieri, Particle Accelerators 60 (1998) 257.

Justifications of this extension

6

A mean (phonon) field theory, same as BCS theory, differs from

Eliashberg.

Consider single photon absorption only, same as M-B theory, differs

from de Visser*.

Dealing with moving Cooper pairs, same as Bardeen**.

The way to deal with mean free path, same as M-B theory.

Calculates the “ideal” case, which means the best performance a

cavity can achieve.

A cavity with short mean free path is not necessary to be a “non-

ideal” cavity.

No extra parameters needed except those in M-B theory + residual

How good it can explain experimental results?

*de Visser, P.J., et al., physical review letters, 2014. 112: p.

047004.**Bardeen, J., Reviews of Modern Physics, 1962. 34(4): p.

667.

7

Cooper pair with zero momentumTwo particles can be attractive to each other with

electron-phonon interaction, if they have:

• momentums in opposite direction

• same energy state k (one ↑ the other ↓)

• k in a shell nearby Fermi level kF, with its

thickness determined by Debye energy

Consequence of attraction:

• Bonded particles become Bosons and get

condensed

• Excited particles obey Fermi-Dirac distribution

• A “forbidden zone” appears nearby kF, with its

thickness to be 2𝜟 (energy gap)

• Energy of excited particle (hole below kF and

electron above kF) changes from|εk|=|1/2mvk2-

εF| before condensation to 𝑬𝒌 = 𝜺𝒌𝟐 + 𝜟𝟐

after condensation

A macroscopic quantum effect!

εF=225meVkF

k↑

-k↓ εk=1/2mvk2- εF

2kTD=47.4me

V

(Energies are based on Nb with selected

parameters)

8

Cooper pair with a net momentum

“States with a net current

flow can be obtained by

taking a pairing (k1↑, k2↓)

with k1+k2 =2q, and 2q the

same for all virtual pairs” – quoted from BCS theory

A small net momentum appears

Consequence:

• Energy split appears for ↑ and ↓

• The energy split is angle dependent

• This angle can be any number

𝒗𝑭

𝒗𝒔α

k+q↑

-k+q↓

2q

2εs<0.0048meV

εk+q=1/2m(𝑣𝑘+ 𝑣𝑠)2 - εF= εk + εs + εext

ε-k+q=1/2m(𝑣𝑘- 𝑣𝑠)2 - εF= εk + εs - εext

εext = mvFvs cos α = pFvsx

9

Energy split caused by moving Cooper pairs

BCS theory extension

Before condensation 𝜀𝑘 𝜀−𝒌+𝒒 = 𝜀𝑘 + 𝜀𝑠 − 𝜀𝑒𝑥𝑡 , 𝜀𝒌+𝒒 = 𝜀𝑘 +

𝜀𝑠 + 𝜀𝑒𝑥𝑡

After condensation 𝐸𝑘 = 𝜀𝒌2 + 𝛥2

1

2 𝐸−𝒌+𝒒↓ = 𝐸𝑘 − 𝜀𝑒𝑥𝑡 , 𝐸𝒌+𝒒↑ = 𝐸𝑘 +

𝜀𝑒𝑥𝑡

with 𝐸𝑘 = 𝜀𝒌 + 𝜀𝑠2 + 𝛥2

1

2

0

1

2

3

4

5

-5 0 5

0

1

2

3

4

5

-6 -4 -2 0 2 4 6𝜺𝒌 𝜺𝒌

𝑬𝒌

𝑬−𝒌+𝒒↓

𝑬𝒌+𝒒↑

|𝜺𝒌|

𝒓𝒆𝒇: 𝜺𝑭

Energy needed to break a Cooper pair,

𝜟 for ↑, and the same for ↓ 𝜟 − 𝜺𝒆𝒙𝒕 for ↑, 𝜟 + 𝜺𝒆𝒙𝒕 for ↓

Energy split

f and h modified by moving Cooper pairs

10

Modified density of states and probability of single occupation at T<Tc:

Low field limit density of states

and distribution function

Below EF Above EF Density of states and distribution

function with moving cooper pairs,

angle averaged

-𝟏. 𝟐∆

Plots with 𝑷𝑭𝑽𝒔= 0.4∆and T/Tc=0.97

Density of states and distribution

function with moving cooper

pairs, angle-dependent

-- α = π

-- α = π/2

-- α = 0

-

𝟐∆

1

𝑁(0)

𝑑𝑁(𝐸)

𝑑𝐸=𝑑𝜀

𝑑𝐸

J. P. Turneaure, J. Halbritter, and H. A.

Schwettman, Journal of Superconductivity 4,

341 (1991)

For electron

For hole

For electron

For hole

𝟏. 𝟐∆Also appeared in I.O. Kulik, V.

Palmieri, Particle Accelerators 60

(1998) 257.

Modification of M-B theory by moving Cooper pairs

Scattering happens between

any two k and k’

11

any two k and k’ with q,

𝑬𝒌 + 𝜺𝒆𝒙𝒕 and 𝑬𝒌′ + 𝜺𝒆𝒙𝒕′

BCS/M-B Extension

Net momentum 0 2q

Initial X0

00

(k↑, -k↓)

(k’↑,-k’↓)

(k+q↑, -k+q↓)

(k’+q↑,-k’+q↓)

Final 00

X0

(k↑, -k↓)

(k’↑,-k’↓)

(k+q↑, -k+q↓)

(k’+q↑,-k’+q↓)

Ground(+) or excited(-) +

+

(k↑, -k↓)

(k’↑,-k’↓)

(k+q↑, -k+q↓)

(k’+q↑,-k’+q↓)

Energy difference Wi-Wf Ek↑-Ek’↑ Ek+q↑-Ek’+q↑

Probability of initial state f dependent Modified f dependent

Scattering matrix elements h dependent Modified h dependent

Absorbing/releasing one photon: additional energy difference ±ℏ(ω-is),

s→0

Final expression

• The final expression is a quadruple integration, besides the

integrations in energy and in reciprocal space shown in M-B

theory, the extension has two additional integrations in angles,

related to k and k’.

• The averaging over both RF cycle and depth into the surface

requires two additional integrations.

• A MathematicaTM script was developed to calculate the Rs vs Bpk.

It is slow, but it works.

• No parameter fittings can be done using current script due to the

slow calculation speed.

12

Calculation result and explanation (1)

13

Surface resistance, Rs, (red line) and reactance, Xs, (blue dashed line)

versus Cooper pair velocity and corresponding magnetic field for Nb at 2 K

and 1.5 GHz.

0 50 100 150

584

588

592

596

600

604

608

0

5

10

15

20

25

30

0 200 400 600 800 1000

Magnetic flux density (mT)

Xs

(µΩ

)

Rs

(nΩ

)

vs (m/s)

*M-B

Why decreasing?

Calculation result and explanation (2)

14

E+ħ𝝎E

𝑹 ∝ [𝒇 𝑬 − 𝒇 𝑬 + ℏω ]𝒈𝒅𝑬

The “golden rule” in extreme anomalous limit

and low temperature approximation

Note that 𝑷𝑭𝑽𝒔>>ħ𝝎 could happen, the overlap

between red and purple could be significant.

Net effect: release energy, cause Rs

𝑹 ∝ [𝒇 𝑬𝟏 − 𝒇 𝑬𝟐 ][𝒇 εext +𝒇 −εext ]𝒈(𝒉)𝒅𝑬

In M-B theory, mathematically, the scattering between

any two k and k’ with photon interaction equals to the

scattering between E and E+ħ𝝎.

With moving Cooper pairs, mathematically, the

scattering between any two k and k’ with photon

interaction equals to the scattering between E+εext and

E+ε’ext+ ħ𝝎.

Rs decreasing?• Source: angle between 𝑽𝑭 (any direction) and 𝑽𝒔 cause

energy split with angle dependence.

• Consequence: Energy split and modified single

particle distribution function cause an overall

reduction effect in scattering.

Any E’Any E

Absorb/release a photon

Net effect: release energy, cause Rs

Absorb a photon

𝑷𝑭𝑽𝒔

E+ε’ext+ ħ𝝎E+εext

E+εext E+ε’ext+ ħ𝝎

Absorb

a

photon

Term 1 Term 2 Term 3

0E+00

1E+10

2E+10

3E+10

4E+10

5E+10

6E+10

7E+10

0 20 40 60 80 100 120

Q0

Bpk (mT)

2.0 K, 1.5 GHz - Theory + 1.7 nohm

G1G2 1400C (LG), 1.5 GHz, 2.0 K

2.0 K, 1.3 GHz - Theory + 3.0 nohm

TE1AES005 800C (FG), 1.3 GHz, 2.0 K

TE1AES003 800C (FG), 1.3 GHz, 2.0 K

Calc for: = 32 nm= 40 nm/Tc = 1.85

mfp = 50 nm

Theory vs Experiments

15

Mattis-Bardeen

P. Dhakal, et al., PRST-AB, 2013. 16(4): p.

042001.

A. Grassellino, et al., Supercon. Sci.and Tech.,

2013. 26(10): p. 102001.

No need for extra parameters!

16

More data…Palczewski et al., LINAC2014

About the inconsistency at

the beginning, there are

several possibilities:

1, The choose of

parameters

2, Measurement errors

3, Cavity performance

could be further improved

4, Some facts that are not

considered in this model:

phonon distribution,

multiple photo absorption,

additional non-linear

effects, etc.

“Textbook” values

We actually predicted the

behavior at low temperatures

Eichhorn et al.

17

Exciting? Let’s be honest

0E+00

5E+09

1E+10

2E+10

2E+10

3E+10

3E+10

4E+10

4E+10

5E+10

5E+10

0 5 10 15 20 25 30 35

+5 microns EP

+7 microns EP

+10 microns EP

A. Grassellino et al.

Dhakal et al.

Summary

18

Previous surface impedance calculations are available only for the

low field limit.

A field-dependent derivation of the Mattis-Bardeen theory of SRF

surface impedance has been developed.

The extended range of gradients is treated for the first time.

Without any extra parameters except those from original M-B theory,

field-dependent Rs agreement with experiment with recent heat-

treated/Nb-doping Nb with unusual surface loading is excellent at

different temperatures, with residual resistance to be constant.

The reduction in resistance with increasing field is seen to be an

intrinsic effect.

For type-I, and type-II under Hc1.

What is going to happen between Hc1 and Hc2?

19

Thank you for your attention!

I would like to thank Drs. Ilan Ben-Zvi and

Sergey Belomestnykh for their comments and

suggestions during the preparation of this talk.